This tool shows UMP and EMP when preferences are satiated: there is a bliss point \( (\delta_x, \delta_y) \) where utility peaks, and utility falls as the bundle moves away from it. Preferences are therefore non-monotone — more is not always better (local nonsatiation fails). When the bliss point is affordable, the consumer always buys it, whatever their income, and leaves unspent any income beyond its cost, so the budget need not bind and Walras’ law fails. These are the definitions of UMP and EMP.
\[ \text{(UMP)}\qquad \max_{(x,y) \in \mathbb{R}^2_+}\; u(x,y) \quad\text{s.t.}\quad p_x x + p_y y \le M \] \[ \text{(EMP)}\qquad \min_{(x,y) \in \mathbb{R}^2_+}\; \bigl(p_x x + p_y y\bigr) \quad\text{s.t.}\quad u(x,y) \ge \bar u \]Choose how distance is measured and a problem, then move the bliss point, prices, and income, and watch the optimal bundle respond.
What to look for
- Lower the prices or raise income until the bliss point is affordable. Does the consumer still always spend all income?
- Make the bliss point unaffordable. Where does the optimum sit relative to the budget line and the bliss point?
- Switch the distance measure between Euclidean and taxicab, and observe how the optimum changes.